Describing the path of a heat seeking particle

[azure kitsune]

Homework Statement

Find the path of a heat-seeking particle placed at (4,3,10) with a temperature field

$$T(x,y,z) = 400 - 2x^2 - y^2 -4z^2$$

Homework Equations

Formulas for directional derivative and gradient.

The Attempt at a Solution

At any point in space (x,y,z), the particle must be moving in the direction which causes the greatest increase in T. This direction is given by the direction of

$$\nabla T = < -4x, -2y, -8z >$$.

I do not know how to continue from here.

 

[HallsofIvy]

The velocity vector points in the direction of $\nabla T$ and so must be some multiple, say "k", of it.

That means you must have dx/dt= -4kx, dy/dt= -2ky, and dz/dt= -8kz for some number k.
You also are given that x(0)= 4(0), y(0)= 3, z(0)= 10. You can solve each of those for x, y, and z in terms of kt. That gives parametric equations for the path with parameter s= kt.